|
| |
|
|
A234253
|
|
a(n) = sum_{i=1..n} C(7+i,8)^2.
|
|
3
|
|
|
|
1, 82, 2107, 29332, 274357, 1930726, 10948735, 52357960, 217994860, 808970960, 2723733524, 8436372248, 24304813148, 65712993248, 167965846148, 408373664744, 949291256585, 2119095737210, 4559798912835, 9488531918460, 19148848609485, 37571357310510
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
In general we have the following formula : a(n) = Sum_{i=1..n}C(e-1+i,e)^2 = C(n+e-1,e+1)*Fe(n)/C(2*e+1). We have the following definition : Fe(n) = Sum_{i=1..n}(-1)^(e+i)*C(e+i,i)*C(n+e,i), and Fe(1) = C(2*e+1,e). [This needs clarification, Joerg Arndt, May 04 2014]
|
|
|
LINKS
|
Index entries for linear recurrences with constant coefficients, signature (18, -153, 816, -3060, 8568, -18564, 31824, -43758, 48620, -43758, 31824, -18564, 8568, -3060, 816, -153, 18, -1).
|
|
|
FORMULA
|
a(n) = sum_{i=1..n)C(7+i,8)^2 = C(n+8,9)*F8(n)/C(17,8) ; F8(n)= Sum_{i=1..n}(-1)^(8+i)*C(8+i,i)*C(n+8,i) = C(8,0)*C(n+8,0) - C(9,1)*C(n+8,1) + C(10,2)*C(n+8,2) - C(11,3)*C(n+8,3) + C(12,4)*C(n+8,4) - C(13,5)*C(n+8,5) + C(14,6)*C(n+8,6) - C(15,7)*C(n+8,7) + C(16,8)*Cn+8,8). We have the following values for F8(n) : F8(0)=1, F8(1)=24310, F8(2)=199342, F8(3)=931294, .... [This needs clarification, Joerg Arndt, May 04 2014]
G.f.: x*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1) / (x-1)^18. - Colin Barker, May 02 2014
|
|
|
EXAMPLE
|
For n=3, Sum_{i=1..3)C(7+i,8)^2 = C(11,9)*F8(3)/C(17,8) = 55*931294/24310 = 2107. [This needs clarification, Joerg Arndt, May 04 2014]
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
Table[Sum[Binomial[7 + i, 8]^2, {i, n}], {n, 30}] (* Wesley Ivan Hurt, Dec 23 2013 *)
CoefficientList[Series[(x^8 + 64 x^7 + 784 x^6 + 3136 x^5 + 4900 x^4 + 3136 x^3 + 784 x^2 + 64 x + 1)/(x - 1)^18, {x, 0, 40}], x] (* Vincenzo Librandi, May 06 2014 *)
|
|
|
PROG
|
(PARI) Vec(x*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1)/(x-1)^18 + O(x^100)) \\ Colin Barker, May 02 2014
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
One term corrected and more terms from Colin Barker, May 02 2014
|
|
|
STATUS
|
approved
|
| |
|
|
